In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1970 (Submitted May JAMES HERBERT STARNES, JR

An experimental and theoretical investigation of the effect of a circular bore on the bending of thin cylindrical shells under axial compression was carried out. The Mylar shells were tested in a controlled load testing machine and showed the effect of increasing the hole radius on the cylinder bending loads. The solution provided an upper limit for the bending stresses of cylinders tested for hole radii less than ten percent of the shell radii.

The theoretical part of the study showed that even a small hole should reduce the buckling stresses significantly. For very small holes, the shell buckled into the general collapse configuration and there was no apparent effect of the hole on the buckling mode of the shell. Tensile loads of shell 7 Tensile loads of shell 14 Tensile loads of shell 17 Tensile loads of shell 20.

Shell Buckling Stresses 6 Shell Buckling Stresses 7 Shell Buckling Stresses 14 Shell Buckling Stresses 17 Shell Buckling Stresses 20. Shell Buckling Loads 6 Shell Buckling Loads 14 Shell Buckling Stresses 20 Shell Buckling Stresses 14 Buckling St res that s of Shell 20.

INTRODUCTION

The average value of the elastic modulus of the seam found in this way was 6. These strain gauges were mounted in pairs every 30 degrees around the circumference of the load cell. The other end of the specimens was attached to the load cell in a similar manner, and then the spacer was attached to the load cell.

Tests were performed to determine the effect of a hole in the shell on the detection signal. This scan provided a measure of the initial imperfections in the shell and provided a reference surface corresponding to the unloaded condition. The first load increase was applied to the shell by rotating the threaded shafts of the test machine.

The distance from the load cell to the edge of the largest hole tested was 4. To prevent premature buckling of the shell, no adjustment was made to the applied load.

RESULTS OF THE EXPERIMENT

ANALYSIS A. DEVELOPMENT OF THE ANALYSIS

It also requires that the local bending stresses in the cavity region are shown to exist at. Lekkerkerker, is assumed to make only a small contribution to the initial cracking of the shell and can be neglected. This assumption is justified by Lekkerkerker's membrane stress results, which approach the flat plate stress concentration values at the hole as the hole becomes small.

To solve the problem, within the constraints of the above assumptions, a coordinate system is adopted which has its origin at the middle surface of the cylinder at the center of the hole as shown in Fig. The assumed displacement function must become zero as the distance from the hole becomes large and is not required to be zero at the hole. The trigonometric form of this function was suggested by the local buckling pattern of the Mylar shell experiments.

Before expressing the final form of the membrane energy, it is necessary to determine the voltage function F. After collecting the equal terms of the undetermined coefficients, equations (23) can be written in matrix form as

RESULTS OF THE ANALYSIS The elements of the matrix B

CONCLUSIONS

As a result of this study, it is possible to conclude that a circular hole in a cylinder can greatly reduce the buckling stresses of the cylinder. The nature of the shell cracking can be described as a local cracking phenomenon leading to the general collapse of the shell. If the hole is small enough, the stress concentration at the hole is not sufficient to cause buckling due to the hole before the shell.

The stability of the local bending mode for larger holes depends on whether the stress level in the shell is high enough to make the shell sensitive to small perturbations. At moderate values of µ, local buckling in the region of the hole causes enough perturbation to cause general collapse of the shell without increasing the applied load. For larger µ values, the local buckling in the bore area is stable, and general collapse occurs only after increasing the applied load.

The results can be extended to include both applied axial loads and bending moments about an axis perpendicular to the cylinder diameter passing through the center of the hole. This would be done by interpreting the relative stresses applied to the cylinder generators tangent to the hole as the applied stress. The simplified analytical approximation presented in this thesis provides a reasonable solution for values of µ less than approximately 2.

For larger values of µ, the problem should be treated as a nonlinear response problem. A nonlinear approach would also be necessary to calculate the general collapse stresses that occur after the occurrence of stable local buckling. The orientation of the local bending pattern of the copper shell experiments indicates that the assumed displacement function used in the analysis may not have been general enough.

Second, the relationship between the hole effect and other initial imperfections was not considered. While the magnitude of these bending stresses is always smaller than the membrane stresses in the hole region, it is very likely that they play some role in local bending. Effects of unreinforced circular cutouts on buckling of circular cylindrical shells under axial compression.

APPENDIX I